# Talk:Preimage of Intersection under Relation

## Equality is true

Equality is true if you consider preimages of a map. In particular all of the following are true:

Let $f$ be a map from $A$ to $B$, where $A$ and $B$ are sets. Then the following are true:

$f^{-1}(A\cap B)=f^{-1}(A)\cap f^{-1}(B)$
$f^{-1}(A\cup B)=f^{-1}(A)\cup f^{-1}(B)$
$f^{-1}(A\setminus B)=f^{-1}(A)\setminus f^{-1}(B)$

In a while I'll try to see if I can split the page Definition:Preimage into pages about maps and relations. Then I'll add pages for all of these, given they don't exist (I haven't yet checked). JSchoone 17:15, 10 December 2011 (CST)

All these are proved in Preimage of Intersection under Mapping and its related pages. The condition under which equality holds for $\cap$ and $\setminus$ are also explored in detail (something rarely seen in the literature). --prime mover 17:51, 10 December 2011 (CST)
Definition:Preimage has now been split appropriately. --prime mover 05:17, 18 March 2012 (EDT)